Hydromagnetic turbulent flow past a semi infinite vertical plate subjected to heat flux

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Hydromagnetic turbulent flow past a semi infinite vertical plate subjected to heat flux

  1. 1. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.8, 2012 Hydromagnetic Turbulent Flow Past A Semi-Infinite Vertical Plate Subjected To Heat Flux Emmah Marigi 1* Matthew Kinyanjui2 Jackson Kwanza3 1. Kimathi University College of Technology P. O. box 657-10100, Nyeri, Kenya 2. Jomo Kenyatta University of Agriculture and Technology P. O. Box 62000, Nairobi, Kenya 3. Jomo Kenyatta University of Agriculture and Technology P. O. Box 62000, Nairobi, Kenya * E-mail of the corresponding author emumarigi@gmail.comAbstractIn this study we have investigated a turbulent flow of a rotating fluid past a semi-infinite vertical porous platesubjected to a constant heat flux. A variable magnetic field is applied transversely in the direction normal to theplate. An induced electric current known as Hall current exists due to the presence of both electric field andmagnetic field. The differential equations governing this problem are solved numerically using a finite differencescheme. Further we have investigated the effects of various parameters on the velocity, temperature, magneticfield and concentration profile. The skin friction the rate of heat transfer and the rate of mass transfer iscalculated using Newton’s interpolation formula We noted that the Hall current, rotation parameter, Eckertnumber, injection and Schmidt number affect the velocity, temperature magnetic field and concentration profiles.Keywords: Free convection, Turbulent flow, rotation, Magnetic field, Hall current, Heat flux Heat Transfer,Mass transfer1 IntroductionIn recent years the theoretical study of MHD flows has been a subject of great interest due to its widely spreadapplication on separation of matter from fluids, designing of cooling systems with liquid metals, petroleumindustry, purification of crude oil, and many other applications. The study of fluids on a rotating system has received considerable interest due to its application in practicalsituations like meteorology, geophysical fluids dynamics, gaseous and nuclear reactions. Debnath et.al. [1]Investigated the unsteady rotating flow of a viscous fluid in the presence of Hall Effect. Hayat et.al. [2]Discussed effect of Hall current and heat transfer on rotating flow on a second grade fluid through a porousmedium. Katagiri [3] discussed the effect of Hall current on the MHD boundary layer flow past a semi-infiniteplate. Kinyanjui et al [4] studied the MHD Stokes problem for a vertical infinite plate in a dissipative rotatingfluid with Hall current.Kinyanjui et al [5] presented their work in MHD free convection heat and mass transfer ofa heat generating fluid past an impulsively started infinite vertical porous plate with Hall current and radiationabsorption. Kwanza J,K, et al. [6] studied Hydromagnetic Free Convection Flow Past a Semi – Infinite VerticalPorous Plate Subjected to Constant Heat Flux With Radiation Absorption.MHD Stokes problem of convectiveflow from a vertical infinite plate in a rotating fluid. was studied by Ram [7]. Soundalgekar, Singh and Takhar [8]study of MHD free convection flow past a semi-infinite vertical plate with suction and injection. The early works on fluid dynamics is mostly on laminar flows with very little devotion to turbulent flowsbut most flows of engineering importance are turbulent and hence the reason for this study. In this study theproblem of MHD free convection turbulent flow of a rotating fluid past a semi-infinite plate is considered.2. Mathematical Analysis.We consider a hydromagnetic turbulent flow of incompressible viscous rotating fluid past an impulsively startedsemi-infinite porous plate. A variable magnetic field H is applied in a direction normal to the plate. The choice ofco-ordinate is such that the y -axis is taken along the plate in the vertical direction and the x -axis is takennormal to the plate. 15
  2. 2. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.8, 2012 Y U0 H0 X Z Figure 1 Geometry of the ProblemIn this case we consider the turbulent flow when the plate is subjected to a constant heat flux. Turbulent flowsare highly irregular and there are rapid fluctuations of velocity in the flow variable with respect to time andlocation. Mean value provides a basis for studying the spatial variation. The instantaneous value for a generalflow say v for a turbulent fluid motion can be given as v = v + v ′ where v is the mean value and v′ thefluctuating component.The Reynolds averaging rules is used to transform equations governing laminar flow to govern turbulent flows.Initially temperature of the fluid and the plate are assumed to be the same. At t > 0, the plate starts movingimpulsively on its plane at a velocity U0 .The plate is maintained at a constant temperature Tw higher than theconstant temperature T∞ of the surrounding fluid and the concentration Cw greater than the constantconcentration C∞ of the surrounding fluid. The flow is turbulent therefore there is a large magnetic fields thisimplies that Hall currents significantly affect the flow. The fluid and the plate are at a state of rigid rotation withuniform angular velocity Ω about the x-axis taken normal to the plate. For a non-zero rotation rate the velocityvector is of the form q = u ( x, y ) % + v ( x, y ) % + w ( x, y ) k . u is the axial velocity while v and w i j %represent the primary and secondary flows. As the rotation is around % it is of the form Ω = Ω % . i iAs the plate is semi-infinite in extent and the flow is unsteady the physical variables are functions ofx∗ , y∗ and t ∗2.1 Governing equationsThe problem is governed by the following set of equations∂v ∗ ∂v ∗ ∂v ∗ ∂ 2v ∗ + u0 ∗ + v ∗ + 2Ωw∗ = υ ∗ 2 + β g (T − T∞ ) + β c ( C ∗ − C∞ ∗ )∂t ∗ ∂x ∂y ∂x (1) ∂ u ∗v∗ µ0 H 0 − + J z∗ ∂x ∗ ρ ∂ w∗ ∂ w∗ ∂ w∗ ∂ 2 w∗ ∂ u ∗ w∗ µ 0 H 0 + u 0 ∗ + v ∗ − 2Ωv = υ ∗ 2 − ∗ − J y∗ (2) ∂t ∗ ∂x ∂y ∂x ∂x ∗ ρ 16
  3. 3. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.8, 2012  ∗  2  ∗  2 ∂T ∗ ∂T ∗ ∂T ∗ κ ∂ 2T ∗ Q∗ ν  ∂ v  +  ∂w   + u0 ∗ + v ∗ ∗ = + + (3)∂t ∗ ∂x ∂y ρ c p ∂x∗2 ρ c p c p  ∂x∗   ∂x∗     ∂H ∗ 1  ∂ 2 H ∗  ∗ ∂H ∗ ∂v ∗ =   −v − H∗ ∗ (4)∂t ∗ µ eσ  ∂x ∗2  ∂y ∗ ∂y ∗∂C ∗ ∂C ∗ ∂C ∂ 2C ∗ + u ∗ ∗ + v∗ ∗ = D ∗ 2 (5)∂t ∗ ∂x ∂y ∂xThe generalized Ohm’s law including the effect of Hall current is written asˆ ωτ ˆ ˆ H ( ˆ  )J + e e J × H = σ  E + µe q × H + ˆ ˆ 1 eη e  ∇Pe   (6)  Solving the above equation for the current density components j y∗ and jz∗ σµe H 0 ( mv∗ + w∗ )J y∗ =ˆ (7) 1 + m2 σµe H 0 ( v∗ − mw∗ )J z∗ = −ˆ (8) 1 + m2Where m = ω eτ e is the Hall parameterSubstituting equations (7) and (8) in (2) and (3) respectively we get∂v ∗ ∂v ∗ ∂v ∗ ∂ 2v ∗ + u0 ∗ + v ∗ + 2Ωw∗ = υ ∗2 + β g (T − T∞ ) + β c g ( C ∗ − C∞∗ )∂t ∗ ∂x ∂y ∂x − ∂u v∗ ∗ − ( ∗ σµe 2 H ∗2 v − mw ∗ ) (9) ∂x∗ ρ (1 + m2 )∂w ∗ ∂w ∂w∗ ∗ ∂ w ∂u w + u0 ∗ + v ∗ − 2Ωv∗ = υ ∗2 − − 2 ∗ ∗ ∗ ( σµe 2 H ∗2 mv + w ∗ ∗ ) ρ (1 + m 2 ) ∗ (10)∂t ∂x ∂y ∂x ∂x∗  ∗  2  ∗  2 ∂T ∗ ∂T ∗ ∂T ∗ κ ∂ 2T ∗ Q ∗ ν  ∂ v  +  ∂ w   + u0 ∗ + v ∗ ∗ = + + (11)∂t ∗ ∂x ∂y ρ c p ∂x ∗ 2 ρ c p c p  ∂x∗   ∂x∗       ∂H ∗ 1  ∂ 2 H ∗  ∗ ∂H ∗ ∂v∗ =   −v − H∗ ∗ (12)∂t ∗ µ eσ  ∂x ∗2  ∂y ∗ ∂y 17
  4. 4. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.8, 2012 ∗∂C ∗ ∗ ∂C ∗ ∗ ∂C ∂ 2C ∗ +u +v = D ∗2 (13)∂t ∗ ∂x∗ ∂y ∗ ∂x2.2 Initial and Boundary conditions.Since heat is constantly being supplied to the plate, we apply Fourier’s law at x ∗ = 0 ∂T ∗ q∗ =− (14) ∂x∗ κt ∗ ≤ 0 : v ∗ ( x∗ , y ∗ , t ∗ ) = 0 w∗ ( x ∗ , y ∗ , t ∗ ) = 0 T ∗ ( x∗ , y ∗ , t ∗ ) = T∞ ∗ (15a) C ∗ ( x ∗ , y ∗ , t ∗ ) = C∞ ∗ ∂T ∗ q∗t ∗ > 0 : v∗ ( 0, y ∗ , t ∗ ) = 0 w∗ ( 0, y ∗ , t ∗ ) = 0 =− C ∗ ( 0, y ∗ , t ∗ ) = Cw∗ (15b) ∂x∗ κ v ∗ ( ∞, y ∗ , t ∗ ) = 0 w∗ ( ∞, y ∗ , t ∗ ) = 0 T ∗ ( ∞, y ∗ , t ∗ ) = T∞ ∗ (15c) C ∗ ( ∞ , y ∗ , t ∗ ) = C∞ ∗In this study non- dimensionalization is based on the following non- dimensional quantities t ∗U 2 x∗U y ∗U u0∗ v∗ w∗t= x= y= u0 = v= w= υ υ υ U U U υ g β  q υ κU  ∗ µcp   U2 H∗Pr = Gr =   Ec = H= κ U3 c p ( q∗υ κU ) H0 T ∗ − T∞ ∗ σµe 2 H 0 2υ Qυθ= M2 = δ= Rm = σµe LU (16)  q ∗υ  ρU 2 κU 2  κU    C∗ − C ∗ υ g β c ( C ∗ − C∞ ∗ ) Ωυ D C = ∗ ∞∗ Gc = Er = Sc = Cw − C∞ U 3 U2 υOn introducing the dimensionless variables in equations (9) to (13) and using ∂vThe boussinesq’s approximations τ = − ρ vw = A which gives ∂z 18
  5. 5. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.8, 2012  ∂v  2uv = − k 2 x 2    ∂x   ∂w  2uw = − k x  2 2  (17)  ∂x We get the final governing equations as∂v ∂v ∂v  ∂ 2v  2  ∂v  2 2  ∂ v   ∂v  2 2 + u0 + v − 2 Er w =  2  + 2k x   + 2k x  2   ∂t ∂x ∂y  ∂x   ∂x   ∂x   ∂x  (18) M 2 H 2 ( v − mw ) + Grθ + Gc C − (1 + m ) 2∂w ∂w ∂w ∂2w  ∂w   ∂ 2 w   ∂w  2 + u0 +v + 2 Er v = 2 + 2k 2 x   + 2k 2 x 2  2   ∂t ∂x ∂y ∂x  ∂x   ∂x   ∂x  (19) M 2 H 2 ( v + mw ) − (1 + m )2∂θ ∂θ ∂θ 1 ∂ 2θ δ  ∂v 2  ∂w 2  + u0 +v = − θ + Q1C + Ec   +   ∂t ∂x ∂y Pr ∂x 2 Pr  ∂x   ∂x     (20)∂H 1 U 2  ∂2 H  ∂H ∂v =  −v −H (21) ∂t Rm υ 2  ∂x 2  ∂y ∂y∂C ∂C ∂C ∂ 2C + + = Sc 2 (22)∂t ∂x ∂y ∂xThe initial and boundary conditions (15a) to (15c) in non-dimensional form becomest ≤ 0 : v ( x, y,0) = 0 w ( x, y,0) = 0 θ ( x, y,0) = 0 C ( x, y,0 ) = 0 (23a) ∂θt > 0 : v ( 0, y , t ) = 1 w ( 0, y , t ) = 1 = −1 C ( 0, y , t ) = 1 (23b) ∂x v ( ∞, y, t ) = 0 w ( ∞, y, t ) = 0 θ ( ∞, y, t ) = 0 C ( ∞, y, t ) = 0 (23c)3 Method of solutionAs it is not possible to get the exact solution for the equations (18) to (22) we generate numerical solutions. Theequations will be solved subject to the boundary conditions (23a, b, c). Writing equations (18) to (22)in finitedifference form similar to that described by kwanza (2010)Computer is used to generate numerical solutions.4 Calculation of the Rate of Heat Transfer and Skin FrictionThe rate of heat transfer is calculated from the temperature distribution in terms of the Nusselt numberThe skin friction is calculated from the velocity profiles using the equations 19
  6. 6. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.8, 2012 ∂v ∂w τ∗ τy = − and τz = − where τ = (24) ρU 2 x=0 x=0 ∂x ∂x These are calculated by numerical differentiation using Newton’s interpolation formula over the first five points 5 Discussions of results A program was run for various values of velocity, temperature and concentration profiles for the finite difference equations. Results are obtained for various values of the parameters m, Ec, Er, U0, and Sc, associated with the governing problem. From Figure 2, we note that: i. An increase in Hall parameter leads to an increase in primary velocity profiles. This is due to the fact that the effective conductivity decreases with the increase in Hall parameter which reduces the magnetic damping force hence the increase in velocity. ii. An increase in the Eckert number Ec leads to an increase in primary velocity profiles. Increase in Ec increases viscous dissipation and therefore decreases the velocity gradient and hence slows down the motion.iii. An increase in the rotation parameter Er leads to a decrease in primary velocity profiles. The increase in frequency of oscillation decreases the thickness of the boundary layer which in effect increases the velocity gradient and hence the velocity decreases.iv. Increase in time increases the primary velocity profiles. With time the flow gets to the free stream and therefore its velocity increases. From Figure 3, we note that: i. An increase in Hall parameter leads to an increase to secondary velocity profiles. This is due to the fact that the effective conductivity decreases with the increase in Hall parameter which reduces the magnetic damping force hence the increase in velocity. ii. An increase in the Ec leads to a decrease in secondary velocity profiles. This is due to increase in viscous dissipation.iii. An increase in the rotation parameter Er leads to a decrease in secondary velocity profiles. The increase in frequency of oscillation decreases the thickness of the boundary layer and hence the velocity decreases.iv. Increase in time decreases the secondary velocity profiles. With time the flow gets to the free stream and therefore its primary velocity diminishes. From Figure 4 we note that: i. An increase in Hall parameter leads to a slight increase in the temperature profiles. Increase in hall parameter increases the thermal boundary layer hence increase the temperature of the flow. ii. An increase in the Ec increases the temperature profiles. This is because increase in Ec reduces the temperature gradient thus increasing the temperature boundary layer and therefore the increase in temperature.iii. An increase in the rotation parameter Er increases the temperature profiles. The frequency of oscillation is increased thus increase the temperature.iv. Increase in time increases the temperature profiles. With time the flow gets to the free stream where the velocity is high the rate of energy transfer is increased and hence increases the temperature. From Figure 5 we note that: i. An increase in Hall current leads to a slight increase in the magnetic field profiles. ii. An increase in the Eckert number Ec increases the magnetic field profiles.iii. An increase in the rotation parameter Er decreases the magnetic field profiles. Increase in Er increases the frequency of oscillationiv. Increase in time increases the magnetic field profiles. From Figure 6 i. An increase in mass diffusion parameter Sc leads to an increase in the concentration profiles. This is because increase in Sc increases molecular diffusivity which results in an increase of the concentration boundary layer. Hence the concentration of the species is higher for large values of Sc. ii. Removal of injection leads to a decrease in the concentration profiles. This is due to the fact that this reduces the growth of the boundary layers and hence the decrease in the concentration profiles.iii. Increase in time increases the concentration profiles. With time the flow gets to the free stream and therefore its concentration increases. From Table 1 we observe that with Gr = 5.0 i. An increase in Hall parameter leads to a decrease in both τ y and τ z .this is due to the fact that increase in Hall parameter increases the velocity and therefore shear resistance is decreased. An increase in Eckert number Ec leads to an decrease in τ y but a slight increase in τ z . This is due to the effect 20
  7. 7. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.8, 2012 of Ec on the velocity which increases the primary velocity but decreases the secondary velocity An increase in Er leads to an increase in both τ y and τ z . Increase in Er decreases the velocity and therefore the shear resistance is increased. An increase in time leads a decrease inτ y but an increase in τ z . in free stream the primary velocity is increased while the secondary velocity is decreased. From table 3 we observe that i. An increase in the Hall parameter leads to an increase in the Nusselt number. This is because Hall parameter increases the velocity of the fluid and this has a direct effect of increasing the rate of heat transfer ii. An increase in Ec , or the rotation parameter Er leads to a decrease in the Nusselt number. This is attributed to the fact that Ec and Er decrease the velocity of the fluid and consequently decrease the rate of heat transfer.iii. An increase in time leads to an increase in the Nusselt number. In the free stream the velocity of the fluid is increased and therefore the rate of heat transfer increases. From table 4 we observe that i. Removal of injection leads to an increase in rate of mass transfer and therefore the rate of mass transfer increases. ii. Increase in mass diffusion parameter Sc, leads an increase in rate of mass transfer. This is because the diffusion is decreased. iii. Increase in time leads to a decrease in rate of mass transfer. With time the flow gets to the free stream where the velocity of the fluid is high and hence the rate of mass transfer. 6 Conclusions In this paper we analyzed the hydromagnetic flow of a rotating fluid past an impulsively started semi-infinite plate. The magnetic field was taken to be variable and we have considered the turbulent boundary layer. The equations governing the hydromantic flow considered are non-linear there we obtained solutions using the finite difference method with uniform mesh system. Stability of the method was tested by choosing smaller time increments which were found to have no significant difference. A numerical method has been using to calculate the skin friction, the rate of heat transfer and the rate of mass transfer. We noted that if heat is supplied to the plate at a constant rate, then the flow field is affected. Due to the strong magnetic field the presence of the hall current affected the flow significantly. In presence of Hall current cooling of the plate by free convection current increases the thermal boundary layer. 7 Validations In the absence of Turbulent and the magnetic field being constant, results agreed with those of Kinyanjui et al (2000) who investigated the MHD free convection heat and mass transfer of a heat generating fluid past an impulsively started infinite vertical porous plate with Hall current and radiation absorption. References Debnath S.C. Ray and Chalterjee. (1979) Effects Of Hall Current On An Unsteady Hydromagnetic Flow Past A Porous Plate In A Rotating Fluid System. Z. Angew ath Mech 59 469-471. Hayat T., Abbus Z. and Asghar S. (2008) Effect Of Hall Current And Heat Transfer On Rotating Flow On A Second Grade Fluid Through A Porous Medium. Communication in non-linear Sci and numerical simulation 13 2177-2192 Katagiri M. (1969) The Effect of Hall Current On The MHD Boundary Layer Flow Past A Semi-Infinite Plate. J. Phys Sos Jpn 27 1051-1059 Kinyanjui M. Charturvedi N and Uppal S.M (1998) MHD Stokes problem for a vertical infinite plate in a dissipative rotating fluid with Hall current Energy Conversion & Management Kinyanjui M, Kwanza J.K., and Uppal S.M. (2001)MHD free convection heat and mass transfer of a heat generating fluid past an impulsively started infinite vertical porous plate with Hall current and radiation absorption. Energy conversion & management Kwanza J,K, Marigi E.M and Kinyanjui M. (2010) Hydromagnetic Free Convection Flow Past a Semi – Infinite Vertical Porous Plate Subjected to Constant Heat Flux With Radiation Absorption. Energy Conversion & Management.51 1326-1332 Ram P.C. and Kinyanjui M and Tarkhar S. (1995) MHD Stokes problem of convective flow from a vertical infinite plate in a rotating fluid Journal of MHD and Plasma research Vol. 5 No. 2/3 pp 91-105 Soundalgekar V.M. Singh M. and Takhar H.S. (1983) MHD free convection flow past a semi-infinite vertical plate with suction and injection. Non-linear analysis methods theory and analysis vol.7 pp941-944 21
  8. 8. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.8, 2012 m Ec Er t I 1.0 0.01 1.0 0.4 II 50.0 0.01 1.0 0.4 III 100.0 0.01 1.0 0.4 IV 1.0 0.02 1.0 0.4 V 1.0 0.05 1.0 0.4 VI 1.0 0.01 1.5 0.4 VII 1.0 0.01 2.0 0.4 VIII 1.0 0.01 1.0 0.2 IX 1.0 0.01 1.0 0.1 Figure 2: Primary Velocity Profile. m Ec Er t I 1.0 0.01 1.0 0.4 II 50.0 0.01 1.0 0.4 III 100.0 0.01 1.0 0.4 IV 1.0 0.02 1.0 0.4 V 1.0 0.05 1.0 0.4 VI 1.0 0.01 1.5 0.4 VII 1.0 0.01 2.0 0.4 VIII 1.0 0.01 1.0 0.2 IX 1.0 0.01 1.0 0.1 Figure 3: secondary Velocity Profile 22
  9. 9. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.8, 2012 m Ec Er t I 1.0 0.01 1.0 0.4 II 50.0 0.01 1.0 0.4 III 100.0 0.01 1.0 0.4 IV 1.0 0.02 1.0 0.4 V 1.0 0.05 1.0 0.4 VI 1.0 0.01 1.5 0.4 VII 1.0 0.01 2.0 0.4 VIII 1.0 0.01 1.0 0.2 IX 1.0 0.01 1.0 0.1 Figure 4: Temperature Profiles m Ec Er t I 1.0 0.01 1.0 0.4 II 50.0 0.01 1.0 0.4 III 100.0 0.01 1.0 0.4 IV 1.0 0.02 1.0 0.4 V 1.0 0.05 1.0 0.4 VI 1.0 0.01 1.5 0.4 VII 1.0 0.01 2.0 0.4 VIII 1.0 0.01 1.0 0.2 IX 1.0 0.01 1.0 0.1 Figure 5: Magnetic field Profile 23
  10. 10. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.8, 2012 U0 Sc t I 0.5 0.4 0.2 II 0.0 0.4 0.2 III 10.0 0.8 0.2 IV 1.0 1.0 0.2 V 1.0 0.4 0.15 VI 1.0 0.4 0.1 Pr = 0.71 M2 = 50 Gr Figure 6: Concentration Profiles.Table 1: Skin Friction τ y and τ z . Pr = 0.7, M2 = 5.0 and Gr = 5.0 (Cooling of the Plate) m Ec Er t τy τz 1.0 0.01 1.0 0.4 1.690614 3.829907 50.0 0.01 1.0 0.4 0.55476 3.474412 100.0 0.01 1.0 0.4 0.548994 3.462327 1.0 0.02 1.0 0.4 1.685874 3.830644 1.0 0.05 1.0 0.4 1.671635 3.832856 1.0 0.01 1.5 0.4 2.728611 5.254831 1.0 0.01 2.0 0.4 4.070204 6.468513 1.0 0.01 1.0 0.2 8.672809 1.433269 1.0 0.01 1.0 0.1 14.11346 0.496905 24
  11. 11. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.8, 2012Table 3: Rate of heat transfer, Nusselt number Nu. m Ec Er t Nu 1.0 0.01 1.0 0.4 -0.66234 50.0 0.01 1.0 0.4 -0.6545 100.0 0.01 1.0 0.4 -0.65447 1.0 0.02 1.0 0.4 -1.35091 1.0 0.05 1.0 0.4 -3.41602 1.0 0.01 1.5 0.4 -0.70427 1.0 0.01 2.0 0.4 -0.75881 1.0 0.01 1.0 0.2 -0.85075 1.0 0.01 1.0 0.1 -0.81331Table 4: Rate of mass transfer, Sh Sc t ShU00.5 0.4 0.2 5.4699690.0 0.4 0.2 8.3336240.5 0.8 0.2 3.7672110.5 1.0 0.2 3.4020.5 0.4 0.15 8.4212130.5 0.4 0.1 14.12577 25
  12. 12. This academic article was published by The International Institute for Science,Technology and Education (IISTE). The IISTE is a pioneer in the Open AccessPublishing service based in the U.S. and Europe. The aim of the institute isAccelerating Global Knowledge Sharing.More information about the publisher can be found in the IISTE’s homepage:http://www.iiste.orgThe IISTE is currently hosting more than 30 peer-reviewed academic journals andcollaborating with academic institutions around the world. Prospective authors ofIISTE journals can find the submission instruction on the following page:http://www.iiste.org/Journals/The IISTE editorial team promises to the review and publish all the qualifiedsubmissions in a fast manner. All the journals articles are available online to thereaders all over the world without financial, legal, or technical barriers other thanthose inseparable from gaining access to the internet itself. Printed version of thejournals is also available upon request of readers and authors.IISTE Knowledge Sharing PartnersEBSCO, Index Copernicus, Ulrichs Periodicals Directory, JournalTOCS, PKP OpenArchives Harvester, Bielefeld Academic Search Engine, ElektronischeZeitschriftenbibliothek EZB, Open J-Gate, OCLC WorldCat, Universe DigtialLibrary , NewJour, Google Scholar

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